Shape of object changes when camera moves?
The problem that I was having was two separate problems:


The ground plane going bonkers in the camera preview
The ground plane appearing to be in the wrong place in renders


The solution to the first problem turned out to be increasing the near clipping distance on my camera from 1mm to 1cm, though I'm not entirely sure why the clipping distance has an affect on geometry in the preview; the shifting geometry also is preview only and is not reflected in the final render (thanks to Duarte in the comments for pointing this out).

The solution to the second problem only seemed to be related to the first, and turned out to be that I had at some point disabled the subsurf modifier on render, causing the rendered geometry to be completely wrong. Re-enabling it fixed the issue completely. Always check your modifiers!